GATE ECE (Electronics) Test: Auto Correlation & Power Spectral Density

A random process is said to be strictly stationary if, for each n, and each choice of t₁,t₂, . . . ,t_n, the joint CDF of X(t₁), X(t₂), . . . , X(t_n) is the same as the joint CDF of X(t₁ +t), X(t₂ + t), . . . , X(t_n +t), for any t. That is, the statistics of the random process are invariant to time shifts.

Strict sense stationarity is a very strong condition to require on a random process. In practice, it is enough if only first and second-order conditions are satisfied.

The noise at the input to an ideal frequency detector is white. The power spectral density of the noise at the output is Parabolic. 

The Auto - correlation function is a measure of similarity between a signal and itself delayed by τ.

The function is given by,

F[x (t)] ↔ x(w)

F[x (t - τ)] ↔ e^-jwτ ×(w).

F[x × (t - τ)] ↔ e-jwτ x × (w).

By using parseval's identity for transform,

Inverse Fourier transform of square of sine function is always a triangular signal in time domain.

White noise is that signal whose frequency spectrum is uniform i.e. it has flat spectral density.

The power spectral density (PSD) of white noise is uniform throughout the frequency spectrum as shown:

The spectral density of white noise is Uniform and the autocorrelation function of White noise is the Delta function.

Derivation:

The power spectral density is basically the Fourier transform of the autocorrelation function of the power signal, i.e.

Also, the inverse Fourier transform of a constant function is a unit impulse.

Now, the auto-correlation is the inverse Fourier transform (IFT) of power spectral density function.

The inverse Fourier transform of the power spectrum of white noise will be an impulse as shown:

White noise is that signal whose frequency spectrum is uniform i.e. it has flat spectral density.

The power spectral density (PSD) of white noise is uniform throughout the frequency spectrum as shown:

In practice, power spectral density is used to quantify random vibration fatigue.

• Power spectral density (PSD) is the energy variation that takes place within a vibrational signal, measured as frequency per unit of mass. In other words, for each frequency, the spectral density function shows whether the energy that is present is higher or lower. 
• Vibration in the real world is often "random" with many different frequency components. Power spectral densities (PSD or, as they are often called, acceleration spectral densities or ASD for vibration) are used to quantify and compare different vibration environments.
• In real-world Power spectral density is used in the packaging industry to measure how vibrations may affect the goods and this helps in quantifying the random vibration fatigue on goods packed in packaging.

Concept:

ACF is defined as:

R_x(τ) = E[x(t) x(t + τ)] = E[x(t) x(t - τ)]

Properties of ACF:

1. R_x(-τ) = R_x (τ)

2. R_x (0) = E [x²(t)] = Power of x(t)

Calculation:

Given:

x(t) = 3 V(t) – 8,  R_v(τ) = 4e^-5|τ|

E [V(t)] = 0

We know that,

Power of x(t) = E[x²(t)]

= E[9V²(t) + 64 – 48 E[V(t)]]

= 9E [V²(t)] + 64 – 48 E[V(t)]

Now,

E[V²(t)] = R_v(0) = 4

So,

Power of x(t) = (9 × 4) + 64 – 0

Power of x(t) = 100  

White noise is that signal whose frequency spectrum is uniform i.e. it has flat spectral density.

The power spectral density (PSD) of white noise is uniform throughout the frequency spectrum as shown:

Concept:

The power spectral density is basically the Fourier transform of the autocorrelation function of the power signal, i.e.

S_x(f) = F.T.{R_x(τ)}

Analysis:

Given, the autocorrelation function of the random signal X(t) as:

R_X(τ) = e^−2|τ|

So, its power spectral density is obtained as:

D is at f = 0,

S_X(0) = 1

For a random process X(t), the Fourier transform of the autocorrelation function R_X(τ ), denoted by S_X(f), is called the power spectral density of process X(t).

Thus 

Some Properties of Power spectral Density:

→ S_X(f) is a real-valued even function, and S_X(f) ≥ 0 for all f.

→The integral over the frequency range - π  to π  is proportional to the variance of a zero-mean random process and 2π  is the proportionality coefficient. 

→ 

R_X(0) represents the average power in a WSS process.